Revising seismic behaviour factors for reinforced concrete bridge design in the longitudinal direction using multi-objective evolutionary algorithms

  • Vítor T. CamachoEmail author
  • Mário Lopes
  • Carlos S. Oliveira
Original Research


Seismic behaviour factors represent the ratio between the strength of a structure, assuming it always maintains an elastic behaviour, and the strength demand with plastic behaviour and consequent loss of stiffness, at the seismic target displacement. This value is closely related to ductility and to energy dissipation due to hysteretic behaviour. The use of behaviour factors allows to design structures with elastic models, without having to explicitly account for material non-linearity while taking advantage of ductility. However, the definition of these values is not easy, and is dependent on several factors. In bridges, these factors can be, among others, regularity of the bridge in terms of pier height, concrete and steel quality, size of elements and amount of steel reinforcement, pier confinement, etc. These factors influence ductility demand and available ductility in different ways and through multi-objective optimization (MOO), the infrastructure solutions that maximize the use of the available ductility under a given earthquake action and for a given bridge superstructure, pier height scheme and ductility class according to Eurocode 8—part 2, can be obtained. Those optimized solutions, which are obtained through the minimization of steel and concrete in the piers as concurrent objectives, are associated with the maximum behaviour factors that can be used in the design of a given bridge and can be compared with the values recommended by EC8—part 2. Without loss of generality, the methodology is applied to a set of case-studies composed of RC bridges with four 30-m spans and circular piers, analysed in the longitudinal direction and without accounting for abutment effects. With the results from the MOO, the behaviour factors associated to solutions with different ductility levels and pier irregularity schemes are calculated and equations are derived, relating the obtained behaviour factors with a pier irregularity measure and ductility level. The results also show the importance of the choice of stiffness used in the design process.


Seismic behaviour factors Multi-objective optimization Evolutionary algorithms Bridge seismic design Non-linear static analysis 



We acknowledge CERIS/DECivil from IST for all the support.


Vítor T. Camacho has a Grant [Grant Number PD/BD/127802/2016] from Fundação para a Ciência e Tecnologia (FCT).


  1. Alam MI, Baburaj K, Prasun J (2019) Optimal design of thin-walled open cross-section column for maximum buckling load. Thin-Walled Struct 141:423–434CrossRefGoogle Scholar
  2. Arellano H, Dante T, Roberto G (2018) Optimum criss crossing cables in multi-span cable-stayed bridges using genetic algorithms. KSCE J Civil Eng 23:719–728CrossRefGoogle Scholar
  3. Azizi M, Ejlali RG, Ghasemi SAM, Talatahari S (2019) Upgraded whale optimization algorithm for fuzzy logic based vibration control of nonlinear steel structure. Eng Struct 192:53–70CrossRefGoogle Scholar
  4. Bybordiani M, Kazemzadeh Azad S (2019) Optimum design of steel braced frames considering dynamic soil-structure interaction. Struct Multidiscip Optim 60:1123–1137CrossRefGoogle Scholar
  5. Calvi G, Priestley MJN, Kowalsky M (2013) Displacement-based seismic design of bridges. Struct Eng Int 23:112–121CrossRefGoogle Scholar
  6. Camacho V., Nuno H, Mário L, Oliveira CS (2019) Optimizing earthquake design of reinforced concrete bridge infrastructures based on evolutionary computation techniques. Struct Multidiscip OptimGoogle Scholar
  7. CEN (2005a) EN 1998-2: Eurocode 8: design of structures for earthquake resistance—Part 2: bridges. CEN–European Committee for Standardisation, BrusselsGoogle Scholar
  8. CEN (2005b) EN1998-1: Eurocode 8: design of structures for earthquake resistance–Part 1: general rules, seismic actions and rules for buildings. CEN–European Committee for Standardisation, BrusselsGoogle Scholar
  9. Chow CK, Shiu YY (2012) A multiobjective evolutionary algorithm that diversifies population by its density. IEEE Trans Evolut Comput 16(2):149–172CrossRefGoogle Scholar
  10. Curadelli O, Amani M (2014) Integrated structure-passive control design of linear structures under seismic excitations. Eng Struct 81:256–264CrossRefGoogle Scholar
  11. Deb Kalyanmoy (2001) Multi-objective optimization using evolutionary algorithms. Wiley, New York, NYGoogle Scholar
  12. Esfandiari MJ, Urgessa GS, Sheikholarefin S, Manshadi SHD (2018) Optimization of reinforced concrete frames subjected to historical time-history loadings using DMPSO algorithm. Struct Multidiscip Optim 58:2119–2134CrossRefGoogle Scholar
  13. Fajfar Peter (2000) A nonlinear analysis method for performance-based seismic design. Earthq Spectra 16–3:573–592CrossRefGoogle Scholar
  14. Fardis M, et al. (2005) Designers’ guide to EN 1998-1 and 1998-5. Eurocode 8: design provisions for earthquake resistant structures. Designers’ Guide to EurocodesGoogle Scholar
  15. Fragiadakis Michalis, Papadrakakis Manolis (2008) Performance-based optimum seismic design of reinforced concrete structures. Earthq Eng Struct Dyn 37(6):825–844CrossRefGoogle Scholar
  16. Geem ZW, Joong HK, Loganathan GV (2001) A new heuristic optimization algorithm: harmony search. Simulation 76(2):60–68CrossRefGoogle Scholar
  17. Goldberg David E (1989) Genetic algorithms in search, optimization and machine learning. addison-Wesley Longman Publishing Co., Inc, Boston, MAGoogle Scholar
  18. Ha M-H, Vu Q-A, Truong V-H (2018) Optimum design of stay cables of steel cable-stayed bridges using nonlinear inelastic analysis and genetic algorithm. Structures 16:288–302CrossRefGoogle Scholar
  19. Kappos AJ (1999) Evaluation of behaviour factors on the basis of ductility and overstrength studies. Eng Struct 21:823–835CrossRefGoogle Scholar
  20. Kappos AJ, Saiidi MS, Aydinoglu MN, Isakovic T (2012) Seismic design and assessment of bridges, inelastic methods of analysis and case studies. Springer, New YorkGoogle Scholar
  21. Kappos AJ, Paraskeva TS, Moschonas IF (2013) Response modification factors for concrete bridges in Europe. J Br Eng 18(12):1328–1335CrossRefGoogle Scholar
  22. Kolias B, Fardis MN, Pecker A, Gulvanessian H (2012) Designers’ guide to Eurocode 8: design of bridges for earthquake resistance. Designers’ Guide to EurocodesGoogle Scholar
  23. Lagaros ND, Papadrakakis M (2007) Robust seismic design optimization of steel structures. Struct Multidiscip Optim 33:457–469CrossRefGoogle Scholar
  24. Lagaros Nikos D, Tsompanakis Yiannis (2007) Intelligent computational paradigms in earthquake engineering. Idea Group Inc, HersheyCrossRefGoogle Scholar
  25. Liu M, Burns SA, Wen YK (2003) Optimal seismic design of steel frame buildings based on life cycle cost considerations. Earthq Eng Struct Dyn 32(9):1313–1332CrossRefGoogle Scholar
  26. Mander JB, Priestley MJ, Park R (1988) Theoretical stress-strain model for confined concrete. J Struct Eng 114(8):1804–1826CrossRefGoogle Scholar
  27. Martínez CA, Curadelli O, Compagnoni ME (2014) Optimal placement of nonlinear hysteretic dampers on planar structures under seismic excitation. Eng Struct 65:89–98CrossRefGoogle Scholar
  28. McKenna Frank, Fenves Gregory (1999) OpenSEES - open system for earthquake engineering simulation. The Regents of the University of California, Berkeley, CAGoogle Scholar
  29. Mergos PE (2017) Optimum seismic design of reinforced concrete frames according to Eurocode 8 and fib Model Code 2010. Earthq Eng Struct Dyn 46(7):1181–1201CrossRefGoogle Scholar
  30. Pipa M (1993) Ductility of reinforced concrete elements under cyclic actions. Influence of reinforcement mechanical characteristics, PhD thesis. Instituto Superior Técnico (IST), LisbonGoogle Scholar
  31. Plevris V, Mitropoulou CC, Lagaros ND (2012) Structural seismic design optimization and earthquake engineering: formulations and applications. IGI Global, HersheyCrossRefGoogle Scholar
  32. Priestley MJN (2003) Myths and fallacies in earthquake engineering, revisited. IUSS Press, PaviaGoogle Scholar
  33. Priestley MJN, Calvi GM, Kowalsky MJ, Powell GH (2008) Displacement-based seismic design of structures. Earthq Spectra 24(2):555CrossRefGoogle Scholar
  34. Rojas Hugo A, Foley Christopher, Pezeshk Shahram (2011) Risk-based seismic design for optimal structural and nonstructural system performance. Earthq Spectra 27(3):857–880CrossRefGoogle Scholar
  35. Tsompanakis Y, Papadrakakis M (2004) Large-scale reliability-based structural optimization. Struct Multidiscip Optim 26(6):429–440CrossRefGoogle Scholar
  36. Wang JQ, Li S, Dezfuli FH, Alam MS (2019) Sensitivity analysis and multi-criteria optimization of SMA cable restrainers for longitudinal seismic protection of isolated simply supported highway bridges. Eng Struct 189:509–522CrossRefGoogle Scholar
  37. Zhang Qingfu, Li Hui (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evolut Comput 11(6):712–731CrossRefGoogle Scholar
  38. Žižmond J, Dolšek M (2016) Evaluation of factors influencing the earthquake-resistant design of reinforced concrete frames according to Eurocode 8. Struct Infrastruct Eng 12(10):1323–1341CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.CERIS, Department of Civil EngineeringInstituto Superior Técnico (IST)LisbonPortugal

Personalised recommendations